Foundations of quantum mechanics

PART 1 Mathematical Foundations Chapter 1 Measure and Integral Chapter 2 The Axioms of Hilbert Space Chapter 3 Linear Functionals and Linear Operators 3—2 Sesquilinear functionals and qu

FOUNDATIONS OF QUANTUM MECHANICS

JOSEF M JAUCH

University of Geneva, Switzerland

A'V

ADDISON-WESLEY PUBLISHING COMPANY

Reading, Ma.csac/:useu.c Menlo Park (ali1thrnia London Don Mi/tv, Ontario

www.pdfgrip.com

This book is in the

ADDISON-WESLEY SERIES IN ADVANCED PHYSICS

Consulting Editor: MORTON HAMERMESH

COPYRIGHT 1968 BY ADDISON-WESLEY PUI3LISIIING COMPANY, INC ALL RIGHTS

RESERVED THIS BOOK, OR PARTS IIIERH)F, MAY NOF HE IN ANY FORM

WITHOUT WRITTEN PERMISSION OFIIIE I'U131.1S11114 I'RINIFI) IN TIlE UNIThD STATES

OF AMERICA PUBLISHED SIMULIANI(RJSLY IN CANAI)A I.IIrnARY (W CONGRESS CATAIXE CARl) No 67-2397K.

This book is an advanced text on elementary quantum mechanics

By "elementary" I designate here the subject matter of nonrelativisticquantum mechanics for the simplest physical systems With the word

"advanced" I refer to the use of modern mathematical tools and the careful

study of difficult questions concerning the physical interpretation of quantum

mechanics.

These questions of interpretation have been a source of difficulties from

the beginning of the theory in the late twenties to the present day Theyhave been the subject of numerous controversies and they continue to worrycontemporary thoughtful students of the subject

In spite of these difficulties, quantum mechanics is indispensable formost modern research in physics For this reason every physicist worth his

salt must know how to use at least the language of quantum mechanics For

many forms of communication, knowledge of the approved usage of the

language may be quite sufficient A deeper understanding of the meaning isthen not absolutely indispensable

The pragmatic tendency of modern research has often obscured the

difference between knowing the usage of a language and understanding thetneaning of its concepts There are many students everywhere who passtheir examinations in quantum mechanics with top grades without reallyunderstanding what it all means Often it is even worse than that Instead

of learning quantum mechanics in parrot-like fashion, they may learn inthis fashion only particular approximation techniques (such as perturbationtheory, Feynman diagrams or dispersion relations), which then lead them

to believe that these useful techniques are identical with the conceptual basis

of the theory This tendency appears in scores of textbooks and is encouraged

by some prominent physicists

This text, on the contrary, is not concerned with applications or

approx-linations, but with the conceptual foundations of quantum mechanics It

is restricted to the general aspects of the nonrelativistic theory Other

funda-mental topics such as scattering theory, quantum statistics and relativistic

(Itlailtuni mechanics will be reserved for subsequent publications

v

under-problems of quantum mechanics.

The book consists of three parts Part I, called Matheniatical

Founda-tions, contains in four chapters a sundry collection of mathematical results,

not usually found in the arsenal of a physicist but indispensable for

under-standing the rest of the book I have taken special care to explain, motivate,

and define the basic concepts and to state the impoitant theorems The

theorems are rarely proved, however Most of the concepts are from

func-tional analysis and algebra For a physicist this part may be useful as a

short introduction to certain mathematical results which are applicable inmany other domains of physics The mathematician will find nothing new

here, and after a glance at the notation can proceed to Chapter 5

In Part 2, called Physical Foundations, I present in an axiomatic form the

basic notions of general quantum mechanics, together with a detailed

analysis of the deep epistemological problems connected with them

The central theme here is the lattice of propositions, an empirically

determined algebraic structure which characterizes the intrinsic physical

properties of a quantum system

Part 3 is devoted to the quantum mechanics of elementary particles

The important new notion which is introduced here is localizability, togetherwith homogeneity and isotropy of the physical space In this part the reader

will finally find the link with the conventional presentation of quantum

mechanics And it is here also that he encounters Planck's constant, which

fixes the scale of the quantum features

Of previous publications those of von Neumann have most strongly

influenced the work presented here There is also considerable overlap with

the book by G Ludwig and with lecture notes by G Mackey In addition

to material available in these and other references, it also contains the results

of recent research on the foundations of quantum mechanics carried out in

Geneva over the past seven years

The presentation uses a more modern mathematical language than iscustomary in textbooks of quantum mechanics There are essentially three

reasons for this:

First of all, I believe that mathematics itself can profit by maintaining

its relations with the development of physical ideas In the past, mathematicshas always renewed itself in contact with nature, and without such contacts

it is doomed to become pure syniholisni of ever-increasing abstraction

Second, physical ideas can he expressed much more forcefully and

clearly if they are presented in the appropriate language The use ot such alanguage will enable us to distinguish more easily the difficulties which we

www.pdfgrip.com

PREFACE vii

might call syntactical from those of interpretation Contrary to a

wide-spread belief, mathematical rigor, appropriately applied, does not necessarilyintroduce complications In physics it means that we replace a traditionaland often antiquated language by a precise but necessarily abstract mathe-matical language, with the result that many physically important notions

formerly shrouded in a fog of words become crystal clear and of surprising

simplicity.

Third, in all properly formulated physical ideas there is an economy ofthought which is beautiful to contemplate I have always been convincedthat this esthetic aspect of a well-expressed physical theory is just as in-dispensable as its agreement with experience Only beauty can lead to that

"passionate sympathetic contemplation" of the marvels of the physical

world which the ancient Greeks expressed with the orphic word "theory."

About 350 problems appear throughout the book Most of them are short exercises designed to reinforce the notions introduced in the text.

Others are more or less obvious supplements of the text There are alsosome deeper problems indicated by an asterisk The latter kind are all

supplied with a reference or a hint

There remains the pleasant task of remembering here the many

col-leagues, collaborators, and students who in one way or another have helped

to shape the content and the form of this book

My first attempts to rethink quantum mechanics were very much stimulated by Prof D Finkelstein of Yeshiva University and Prof D.

Speiser of the University of Louvain It was during the year 1958, when all

three of us were spending a very stimulating year at CERN, that we beganexamining the question of possible generalizations of quantum mechanics

Many of the ideas conceived during this time were subsequently elaborated

in publications of my students at the University of Geneva I should mentionhere especially the work of G Emch, M Guenin, J P Marchand, B Misra,and C Piron.

In the early stages I profited much from various discussions and

c9rre-sponcknce with Professor G Mackey Many colleagues have read and

criticized different portions of the manuscript I mention here especially

l)r R Hagedorn of CERN, whose severe criticism of the pedagogical

aspects of the first four chapters has been most valuable With Dr J Bell

ot CERN, I debated especially the sections on hidden variables and the

measuring process The chapter on the measuring process has also beeniiitluenced by correspondence with Prof E Wigner of Princeton and withProt L. Rosenfeld of Copenhagen Several other sections were improved

hy criticism from Prof R Ascoli of Palermo, Prof F Rohrlich of Syracuse,

'iurk Prof C F v Weizsückcr of IIaniburg read and commented on the

section of Chapter 5 Professor (, Mackey read critically

viii PREFACE

the entire manuscript and suggested many improvements I)r C Piroti

and Mr A Salah read the proofs, and I am much indebted to them For their

conscientious and patient collaboration To all of them, and ninny others

too numerous to mention, I wish to express here my thanks

The major portion of the book was written while I had the privilege of

holding an invited professorship at the University of California in Los

Angeles, during the winter semester of 1964 May Professors D Saxon and

R Finkelstein find here my deeply felt gratitude for making this sojourn,and thereby this book, possible The University of Geneva contributed its

share by granting a leave of absence which liberated me from teaching dutiesfor three months

I have also enjoyed the active support of CERN, which, by according

me the status of visiting scientist, has greatly facilitated my access to its

excellent research facilities and contact with numerous other physicists

interested in the matters treated in this book

It is unavoidable that my interpretation of controversial questions isnot shared by all of my correspondents Of course, I alone am responsible

for the answers to such questions which appear in this book

Mrs Dorothy Pederson of Los Angeles and Mlle Frances Prost ofGeneva gave generously of their competent services in typing a difficult

manuscript May they, too, as well as their collaborators, find here my

expression of gratitude

August 1966

www.pdfgrip.com

PART 1 Mathematical Foundations

Chapter 1 Measure and Integral

Chapter 2 The Axioms of Hilbert Space

Chapter 3 Linear Functionals and Linear Operators

3—2 Sesquilinear functionals and quadratic forms 32

Chapter 4 Spectral Theorem and Spectral Representation

4—i Self-adjoint operators in finite-dimensional spaces 47

4—5 Spectral densities and generating vectors 58

ix

x CONTENTS

PART 2 Physical Foundations

Chapter 5 The Propositional Calculus

5—4 Classical systems and Boolean lattices 78

5—5 Compatible and incompatible propositions 80

Chapter 6 States and Observables

6—7 The functional calculus for observables 101

7—4 Alternative ways of introducing hidden variables 119

Chapter 8 Proposition Systems and Projective Geometries

8—3 The structure of irreducible proposition systems 127

8—4 Orthocomplementation and the metric of the vector space 129

Chapter 9 Symmetries and Groups

CONTENTS Xi Chapter 10 The Dynamical Structure

10—1 The time evolution of a system

10—2 The dynamical group

10—3 Different descriptions of the time evolution

10—4 Nonconservative systems

11—1 Uncertainty relations

11—2 General description of the measuring process

11—3 Description of the measuring process for quantum-mechanical

systems 11—4 Properties of the measuring device

11—5 Equivalent states

11—6 Events and data

11—7 Mathematical interlude: The tensor product

11—8 The union and separation of systems

11—9 A model of the measuring process

11—10 Three paradoxes

PART 3 Elementary Particles

Chapter 12 The Elementary Particle in One Dimension

12—1 Localizability

12—2 Homogeneity

12—3 The canonical commutation rules

12—4 The elementary particle

12—5 Velocity and Galilei invariance

12—6 The harmonic oscillator

12—7 A Hilbert space of analytical functions

12—8 Localizability and modularity

1 3—1 Localizability

13—2 Homogeneity and isotropy

13—3 Rotations as kinematical symmetries

13—4 Velocity and Galilei invariance

13—5 Gauge transformations and gauge invariance

13—6 Density and current of an observable

Spin and orbital angular momentum

Spin under space reflection and time inversion Spin in an external force field

Elementary particle with arbitrary spin

Chapter 15 Identical Particles

15—1 Assembly of several particles • • • • • • • • • 269

15—2 Mathematical digression: The multiple tensor product • • • 273

15—3 The notion of identity in quantum mechanics • • • • • 275

15—4 Systems of several identical particles • • • • • • • • 278

www.pdfgrip.com

PART 1 Mathematical Foundations

CHAPTER 1

MEASURE AND INTEGRAL

Therefore there is no perfect measure of continuous quantity except by means

of indivisible continuous quantity, for example by means of a point, and noquantity can be perfectly measured unless it is known how many individualpoints it contains And since these are infinite, therefore their number cannot

be known by a creature but by God alone, who disposes everything in number,weight, and measure

ROBERT GROSSETESTE,13th century A.D

The purpose of this chapter is to acquaint the reader with the modern theory

of integration Section 1-1 contains some basic notions of set theory together

with a list of terms and formulas In Section 1-2 we present the notion of

measure space and some properties of measures We define measures on

c-rings of a class of measurable sets, but we pay no attention to the maximalextensions of such measures The following section (1-3) introduces the measurable and integrable functions and defines the notion of integral.

Section 1-4 introduces the theorem of Radon-Nikodym by way of a trivialexample The last section (1-5) on function spaces forms the bridge to thegeneral theory of Hilbert space to be presented in Chapter 2

1-1 SOME NOTIONS AND NOTATIONS FROM SET THEORY

A collection of objects taken as a whole is called a set The objects which

make up a set are called the elements of the set We denote sets by capital

letters, for instance A, B, , S; and the elements by small letters, for instance

a, b, , x. If the element x is contained in the set S we write x e 5; if it

is not contained in the set S we write x S. If every element of a set A iscontained in B we write A a B or B A, and we say A is a subset of B

If A c B and B a A, then we say the two sets are equal and we write

A=B.

The set A u B denotes the set of elements which are either in A or in B

or in both. It will be called the union of A and B

I

4 MEASURE AND INTEGRAL I-i

Hg 1—1 Relations between point

(b) A n B = 0 (disjoint sets).

sets: (a) A a B (A subset of B);

The set A n B denotes the set of elements which are in A as well as in B

It is called the intersection of A and B

If A is a subset of a set 5, we define by A' (with respect to 5) the set ofall elements which are in S but not in A The set A n B' = A — B is calledthe difference of A and B, or the relative complement of B in A

A subset of a general set S can be defined by a certain property m(x)

The set A of all elements which have property m(x) is written

A = {x :

It means: A is the set of all elements which satisfy property m(x) Thus for

instance the operation A u B can be defined by

Similarly we write

AuBr={x:xeAand/orxeB}.

A nB = {x : xeA and xe B}.

(a)

If there exists no element which has property m(x), then the set

0 = {x : m(x)} defines the empty set We have always, for any set A c 5:

If two sets A and B are such that A n B = 0, they are called disjoint

All these notions can be easily illustrated and remembered by using

point sets in a plane (see Figs 1—1 and 1—2)

We shall now go a step further and consider collections of subsets of aset We speak then of a class of subsets Of particular interest are classes

which are closed with respect to certain operations defined above

Fig 1—2 Intersection-, union-, and difference-sets: (a) A n B;

(b) u B; (c) A B A n B' (for S the entire plane).

www.pdfgrip.com

1-1 SOME NOTIONS AND NOTATIONS FROM SET THEORY 5

The most useful notion is that of a ring A nonempty class of subsets is

called a ring B? if, forAeB? and BeG?, it follows that A uBePA and

A — B e B? Examples of rings are easily constructed One of the simplestpossible is the ring consisting of an arbitrary subset A a S, together with

the sets 0, A' and S

Since A — A = 0, every ring contains the empty set One proves by

mathematical induction that if' A1, A2, , is a finite collection of sets in B?, then

Here we have introduced the easily understandable notation

(JA1 A1 u A2 u u

A a ring with the additional property that, for every countable

sequence A1 (i = 1, 2, . ) of sets contained in B?, we have

1=1

A ring is called an algebra (or Boolean algebra) if it contains 5, or

equiva-lently if A e B? implies A' e B?

The primary purpose for introducing the notions of ring, c-ring, and

algebra of sets is to obtain sufficiently large classes of sets to be useful for a

theory of integration On the other hand, for the construction of a measure,

the class of sets must be restricted so that an explicit construction of a

measure is possible This class must contain certain simple sets and for this

reason we want to construct c-rings of sets generated by a certain class of

subsets How this is done is now to be explained

11 e is any class of sets, we may define a unique ring called the ringgenerated by 1, denoted by G?(t) It is defined as follows: Denote by B?,

(1 e I = some index set) the family of all the rings which contain the class g.The intersection of all is again a ring (Problem 9) and it defines the ring

generated by 1:

B?(t) = fl

It is clearly the smallest ring which contains the class S of sets Furthermore

it is unique The same procedure can be used for c-rings

We are now prepared for the most important notion of this section,

the Borel set

Let Sbe the reallineS= {x cxc +cc}

6 MEASURE AND INTEGRAL 1-1

For e we choose the set of all bounded semiclosed intervals of the form

[a, b) e {x a x 'C b}

The Borel sets on the real line are the sets contained in the c-ring G?(4')

generated by this class e

The choice of semiclosed intervals as starting sets might be somewhat

surprising, but there is a technical reason for this The finite unions of open intervals are a ring (Problem 7), while this is not so for closed or open

semi-intervals. It is, however, a posteriori possible to show that the c-ring

gener-ated by the open or the closed interval is also the class of Borel sets These

properties are not difficult to prove, but they require certain technical devices

which transcend the purpose of this book ([1], §15) We shall therefore

state them without proof:

1) The class of all Borel sets is the c-ring generated by all open or all

closed sets

2) The entire set S is a Borel set The c-ring of Borel sets is thus an algebra.3) Every countable set is a Borel set

The first of these properties permits an extension of the notion of Borel sets

to certain topological spaces For instance, in a locally compact Hausdorif

space one defines the Borel sets as the c-ring generated by all closed subsets[1, Chapter 10]

With this general notion one can define Borel sets, for instance, on an

n-dimensional Euclidean space, on a finite-dimensional manifold, such as a

circle, a torus, or a sphere, and on many other much more complicated

spaces. In many applications which we shall use we have to deal with theBorel sets for an arbitrary closed subset of the real line

6 The class of all subsets of a set S is a ring

7 LetS = {x : < x < -(-x}; then the class of all subsetsofS of the form

is a ring.

www.pdfgrip.com

1-2 THE MEASURE SPACE 7

8 If a ring B? of subsets of S contains 5, then A e B? implies A' e B?, and viceversa.

9 The intersection of any family of rings (a-rings) B?, is a ring (a-ring).

1-2 THE MEASURE SPACE

A measure space is a set of elements S, together with a c-ring M of subsets

of S and a nonnegative function 4u(A), defined on all subsets of the class M,which satisfies certain properties to be enumerated below

The subsets of the c-ring M are called the measurable sets We denote

the measure space by (5, M, ji). Sometimes the explicit reference to themeasurable sets and the measure p is suppressed, and we then simply refer

to S as a measure space Because M is a ring, 0 e M and so the null set is

always measurable It is always possible to arrange that S e M, too, so

Property 1 may be relaxed to include infinite measures Then we can only

require 0 p(A) cc (Most applications, however, will be for finite

measures.) A set function which satisfies property (3) is called c-additive

The sets A e M with p(A) = 0 are called the sets of measure zero A

property which is true for all x e S except on a set of measure zero is said

to be true "almost everywhere" (abbreviated a.e.)

Naturally the question arises whether any measures exist and what theirproperties are Instead of entering into these rather difficult questions, we

shall explicitly exhibit two types of measures which we shall use constantly:the Lebesgue measure and the Lebesgue-Stieltjes measure

For the Lebesgue measure, the c-ring of measurable sets consists of theBorel sets on the real line On the half-open intervals [a, b), with a b, wedefine

p{[a, b)} = b — a.

In the theory of measure, one proves that this set function on the half-openintervals has a unique extension to the Borel sets such that it satisfies con-

ditions (I), (2), and (3) This extension will be called the Lebesgue measure

on the real line (Actually the measure can be further extended to the class

of Lebesgue measurable sets, but we shall not need this extension explicitly.)

8 MEASURE AND INTEGRAL 1-2

The Lebesgue-Stieltjes measure is a generalization of the Lebesgue measure obtained in the following way: Let p(2) be a real valued, non-

decreasing function, defined for — cc A + and such that

p(A+O)= lim p(A +c)= p(A).

+ 0

For any semiopen interval [a, b), (a b), we define

p{[a,b)} =p(b)—p(a).

The unique extension of this set function to the Borel sets on the real line

is called the Lebesgue-Stieltjes measure on the real line

For p(A) = A this measure reduces to the Lebesgue measure The greatergenerality of the Lebesgue-Stieltjes measure is especially convenient for dis-crete measures

We obtain a discrete measure by letting p(A) be constant except for a

countable number of discontinuities 2k' where

P(Ak) = PR — 0) +

The measure is then said to be concentrated at the points 2k with the

weights

We say two measures and P2 are comparable if they are defined on

the same c-ring of measurable sets Thus all measures on the Borel sets

are comparable

In the following we shall examine the relation between different

com-parable measures The main point is the observation that comparable

measures can be partially ordered In the following we shall assume all

measures to be comparable without repeating it

A measure Ri is said to be inferior to a measure P2 if all sets of p2-measure

zero are also of p1-measure zero Pi is also called absolutely continuouswith respect to We use the notation Ri -< Thus Ri -< P2 if and only

if p2(A) = 0 implies p1(A) = 0. Two measures Pi' P2 are said to be equivalent

if both

Pi P2 and we note that it is an equivalence relation (Problem 2)

Two measures Pi and P2 are said to be mutually singular if there exist

two disjoint sets A and B such that A u B = S and such that, for every

measurable set X c 5,

p1(AnX)=p2(BnX)=O.

Examples of mutually singular measures are easily constructed (Problem 8)

If a measure p is absolutely continuous with respect to Lehesgue measure,

it is simply called absolutely continuous

www.pdfgrip.com

1-3 MEASURABLE AND INTEGRABLE FUNCTIONS 9

PROBLEMS

1 Let ji be a discrete Lebesgue-Stie!tjes measure For every Bore! set A, =

Pk1 where the sum extends over a!! Ak, e A.

2 The re!ation —' is an equiva!ence re!ation; this means it is reflexive,

sym-metrica!, and transitive:

(c) r-' ,a2 and r-' imp!ies —'

3 Two discrete measures and on the rea! !ine are equiva!ent if and on!y ifthey are concentrated in the same points and their respective weights are non-zero.

4 Every Lebesgue-Stie!tjes measure on the rea! !ine can be decomposed unique!y

into a discrete part and a continuous part, corresponding to the decomposition

of the nondecreasing function p(A) = pa(A) + into a discrete and a tinuous function The discrete part is constant except on a finite orcountab!y infinite set of points where pa(A) is discontinuous.

con-Every continuous nondecreasing function can be decomposed into anabso!ute!y continuous and a singu!ar function = pa(A) + p8(A) The

function p8(A) is singu!ar in the sense that its derivative p8'(A) exists a!mosteverywhere and is equa! to zero, yet p5(A) is continuous and nondecreasing

([2], Section 25).

*6 Theorem (Lebesgue) A finite nondecreasing function p(A) (or, more generally,

a function of bounded variation) possesses a finite derivative a.e ([2], Section 4).

Theorem (Lebesgue) The necessary and sufficient condition that a finite, tinuous and nondecreasing function is equal to the integral of' its derivative is

con-that it is absolutely continuous ([2], Section 25).

8 Let be a discrete measure concentrated at the points AS" and anotherdiscrete measure concentrated at the points Then the two measures aresingu!ar with respect to one another if and on!y if for a!! pairs of

indices i and k.

'[he theory of measure spaces permits a definition of the integral of functions

which is much more general than the so-called Riemann integral usually

introduced in elementary calculus This more general type of integral, to

he defined now, is absolutely indispensable for the definition of Hilbert spaceand other function spaces used constantly in quantum mechanics

We start with the definition of a function A function f is a

corre-spondence between the elements of a set D1, called the domain of f, and a

set A,-, called the range off such that to every x e D1 there corresponds

exactly one element 1(v) e A1 The elements of 0,- are called the argument

of the function

10 MEASURE AND INTEGRAL 1-3

We remark especially that we define here what is sometimes called asingle-valued function It is possible (and, for a systematic exposition, ad-visable) to treat multivalued functions by reducing them to single-valuedones In analytic function theory this procedure leads in a natural manner

to the theory of Riemann surfaces

We emphasize, too, that a function has three determining elements:

A domain, a range, and a rule of correspondence x —, f(x). It will sometimes

be necessary to distinguish two functions which have different domains,

although in a common part of these domains the value of the two functionsmay agree

B

x

Fig 1—3 The inverse of a function f(x)

The sets D1 and A1 may be quite general sets—for instance, subsets ofreal or complex numbers In that case we obtain real or complex functions

But more often they will consist of points in a topological space, functions

in a function space, or even subsets of sets In the latter case we speak also

of set functions

An example of a set function of great importance is the following:

Let B be a subset of the range A1; we define the inverse image (B)

(see Fig 1—3) by setting

= y for all y e A1

t Note that the inverse image is not the inverse function As a function it is

defined on the subsets of and its values are subsets of 1),.

f(x)

www.pdfgrip.com

1-3 MEASURABLE AND INTEGRABLE FUNCTIONS 11

Although, strictly speaking, these two identity functions should be

dis-tinguished (since in one case the domain is D1, in the other A1), they areusually considered identical, an assumption which is entirely correct only

if D1= A1.

For the rest of this section we shall consider real-valued functions over

a measure space A1 is then a subset of the real line B?

Let (5, M, p) be a measure space and f a real-valued function with

domain D1 = S. We call the function f measurable on S if for every Borelset B on the real line, the setf1(B) is measurable

The simplest examples of measurable functions are obtained from the

set functions XAX) defined by

(1 forxeA,

XA(X)

A set function is measurable if and only if the set A is measurable Indeed

we find immediately that

XA (B)=10 ifl*B,

so that XAX) is measurable if A e M

There is a resemblance between measurable functions in a measure space

and continuous functions in a topological space S A topological space isdefined by the class of all open subsets of S A function f(x) from S onto

A1 e B? is then sajd to be continuous if the inverse image f -1(B) of any open

set is an open set in S One obtains the more general class of measurable

functions by replacing the word "open" by "measurable," in the above

definition of continuous function The class is more general because (at

least in all the measure spaces which we consider) the open sets are able sets One of the most important problems in measure theory is to

measur-identify the class of measurable functions over a measure space An efficientway of doing this is to construct the measurable functions from certain simple

functions by the operations of sums, products, and the passage to the limit

In what follows we shall describe this process, without giving proofs

First, one observes that if two functions f and g are measurable, then

12 MEASURE AND INTEGRAL 1-3

with ; real constants and A, e M, are also measurable. We define the

integral of a simple function as

It is a finite number since the ji(A1) are all finite (we admit only finite

measures).

Next we consider sequences of functions (n = 1, 2, . .). We define

convergence in the measure of such a sequence to a limit function f if, forevery e> 0,

n -.

The notion of measurability is more general than that of integrability, the

latter being restricted by the condition that there exist a finite-valued integral.Since simple functions are not only measurable but also integrable, we can

define the integrable functions as follows: A finite-valued function f on ameasure space (S, M, p) is integrable if there exists a sequence of simple

functions f,, such that f,, tends to f in the measure It follows then that the

numbers = J d4u tend to a limit which defines the integral of f:

The integral defined here corresponds to what in the elementary

inte-gration theory is called the definite integral There is also a concept which

generalizes the usual notion of the indefinite integral In the usual definition,the indefinite integral depends on the lower and upper limit of the integration

variable. It is thus an interval function

More generally we may define a set function v(A) for all A e M by

setting

v(A)

www.pdfgrip.com

1-3 MEASURABLE AND INTEGRABLE FUNCTIONS 13

If the integrable functionf is positive, the set function v(A) is a new measure

defined on the c-ring M It is easy to verify that the two measures v and y

are equivalent (The converse of this is an important theorem which we

shall discuss in the next section.)

If the measure space S is the real line, M are the Borel sets, and p the

Lebesgue measure, then the integral is called the Lebesgue integral We

write for it

J f(x) dx.

Some slight modifications are necessary in some of the definitions and

theorems quoted above, since Lebesgue measure is not a finite measure

Similarly we obtain the Lebesgue-Stieltjes integral if we define the

measure corresponding to some nondecreasing function p(x) We write for

this integral

j fd4u(x).

The notion of measurable and integrable functions is easily extended tocomplex-valued functions by defining such a function as integrable if bothreal and imaginary parts are integrable The integral of the function f =

t + 1J is then defined by

+

PROBLEMS

I A continuous function is measurable

2 There exist measurable functions which are not continuous

3 A function has an inverse if and only if f(x1) =f(x2) implies x1 = x2.

4. If two real-valued functions and g are integrable, then ef, for a constant c,

is integrable and (f + g) is integrable.

5 1ff is integrable, then the positive and negative parts off defined by

and are also integrable.

6 If the real-valued function f is integrable, then the indefinite integral off defines

a finite measure on the class of all measurable sets.

7 The measure of Problem 6 is absolutely continuous with respect to the measure which is used for the definition of the integral.

14 MEASURE AND INTEGRAL 1-4

1-4 THE THEOREM OF RADON-NIKODYM

En this section we shall consider the relations between comparable measures

on a fixed measure space This relation is of the greatest importance in the

applications of measure theory We shall neither give the most general form

of the theorem nor prove it; but we shall illustrate it with an example whichrenders it sufficiently plausible

Let us begin with the example: Let S consist of all the integers, S =

1, 2, , n, , and let p,, be a set of positive numbers such that

The c-ring of measurable sets consists of all the subsets of 5, and the measure

Thus we have established, in this particular case, that if v -< p, there exists

a nonnegative measurable function f on S such that for every measurable

set A we have

v(A) = fcl4u.

.JA

In the special case which we have discussed, the function f is not only

measur-able but also integrmeasur-able; but this is the case if and only if the measure v is

finite, as one can easily verify The generalization of this property to anymeasure is the content of the following theorem

Theorem (Radon-Nikodym): The necessary and sz4fJicient condition that

the measure on the real line S he absolutely continuous with re.spect to

the measure v is that there exists a uniquely determined hounded non—

www.pdfgrip.com

1-4 THE THEOREM OF RADON-NIKODYM 15

negative measurable function f(x) with domain S such that

.JA

If, furthermore, the two measures v and p are equivalent, then the function

f is positive a.e., and

p(A)

= IA!

Ifboth measures are finite then both f and 1/fare integrable

The function f is called the Radon-Nikodym derivative and it is often

written as f = dv/d4u. The notation underlines the analogy of this conceptwith the ordinary derivative of a nondecreasing function The analogy isfurther enhanced by considering the Lebesgue-Stieltjes measure on the real

line determined by a nondecreasing function p(x) If c(x) is another measure

of this kind, then the function f of the Radon-Nikodym theorem is given by

dpf(x) =

If c(x) = x, the measure induced by it is the Lebesgue measure, and we

may write f(x) = dp/dx. In this case the function f(x) coincides (a.e.) with

the usual derivative of p(x)

gdv.

.JA

Furthermore, f(n) I/g(n). If and v are finite measures, both functions f

and p are integrabic.

16 MEASURE AND INTEGRAL 1-5

3 If are three measures such that -< -< ji3, then

djzi djz2 — djzi djz2 djz3 —

We consider the set of all complex, integrable functions on a measure space(S, M, and denote it by L1 1ff e L1 then any scalar multiple cf is also

in L1. Likewise, if f1 and f2 e L1 then f1 + f2 e L1 The functions of the

class L1 are thus a linear man()fold

For any f e L1 we define the norm as

MfM

and for any pair f, g e L1 we define a distance function p(f, g) by setting

p(f,g)= j—gM

Two functions f and

set of measure zero

classes of equivalent

by the distance functions p(f, g)

An important property of the space L1 is its completeness

said to be complete if every fundamental

there exists an integrable function f such that

for

Of more importance in quantum mechanics is the space L2(S, M, p) L2,

which consists of all those measurable functions on S which are

square-integrable Thus f e if

Just as in L', we can define a distance function p(f, g) =

A more general concept is the scalar product (f, g) defined for any pair

of functions f and g in L2 It is defined by the formula

g) = ff*g

wheref* is the complex conjugate of f With this definition we have

1-5 FUNCTION SPACES

g for which p(f, g) = 0 are equal except possibly on a

Two such functions are said to be equivalent Thefunctions form a metric space, with the metric defined

A space isThis means

n, m —* cc,

n —* cc.

www.pdfgrip.com

REFERENCES 17

The space L2 is also complete Thus if is a fundamental sequence, thereexists a functionf such that — —, 0. The space L2 is the basic mathe-

matical object for quantum mechanics and we shall devote the next three

chapters to its study

1 P R HALMO5, Measure Theory Princeton: Van Nostrand (1950).

2 F RIEsz AND B SZ.-NAGY, Functional Analysis New York: F Ungar Publ Co.

(1955).

3 N DUNFORD AND J T SCHWARTZ, Linear Operators (Part I, especially Chapter

III) New York: Ilnterscience Publishers (1958).

CHAPTER 2

THE AXIOMS OF HILBERT SPACE

I think we may safely say that the studies preliminary to the construction of a

great theory should be at least as deliberate and thorough as those that are

preliminary to the building of a dwelling-house

CHARLES S PIERCE

En this chapter we introduce the basic properties of Hubert space in matic form The axioms are given in four groups in Section 2-1 The com-ments on the axioms in Section 2-2 introduce such basic material as linearmanifolds, dimension, Schwartz's and Minkowski's inequalities, strong andweak convergence, and orthonormal systems In Section 2-3 we discuss

axio-various realizations of the abstract Hilbert space We devote the whole ofSection 2-4 to the important distinction between manifolds and subspaces

We also discuss the decomposition theorem with respect to a subspace in

a final section (2-5) we introduce the notion of the lattice of subspaces,

a notion which will play a fundamental role throughout this book

2-1 THE AXIOMS OF HILBERT SPACE

The abstract Hilbert space t is a collection of objects called vectors, denoted

byf g, , which satisfy certain axioms to be enumerated below.

The axioms fall into four groups, each of which refers to a different

structure property of Hilbert space Group 1 expresses the fact that t is alinear vector space over the field of complex numbers Group 2 defines thesea/ar product and the metric Group 3 expresses separability, and group 4,co,npleteness, of the space

I. *' is a linear vector space with complex coefficients This means that toevery pair of elements f g e *" there is associated a third (J + g) e t.Furthermore to every element and every complex number A there corresponds

IS

www.pdfgrip.com

2-2 COMMENTS ON THE AXIOMS 19another element Af e The following rules are postulated:

3 The space is separable This means that there exists a e Jt

(n = 1, 2,. ) with the property that it is dense in in the following sense:

For any f e and any a > 0, there exists at least one element f,, of the

sequence such that

The axioms of groups 1 and 2 describe the Hilbert space as a linear vectorspace with scalar product The special choice of the complex numbers forthc field of the coeflicicnts will he justified later from a physical point of

20 THE AXIOMS OF HILBERT SPACE 2-2

view. Here we remark that one can define Hilbert spaces over the reals and

the quaternions (or any other field), and they share many of the properties

which one finds for the Hilbert space with complex numbers

In the axioms of group 2 we require positive definiteness of the scalar

product There is no difficulty in defining spaces with indefinite scalar

prod-ucts We shall, however, need only the definite scalar product This too

will be justified from a physical point of view

Axioms 3 and 4 are restrictions on the size of the space in opposite

directions. Here one should say that axiom 4 is in some sense superfluous

since it can always be fulfilled by a standard procedure, called the completion

of the space. It is the same kind of procedure used in the construction of the

real numbers from a dense subset such as the rational numbers It is the

axiom which permits the notion of continuity in Hilbert space

Axiom 3, on the other hand, is an important restriction on the size of

the space If it is omitted, one obtains nonseparable spaces These will not

be used in this book since their physical meaning is not yet understood,although many of the properties of separable spaces can be transferred to

the nonseparable ones

The reader may have noticed the absence of a dimension axiom Thisaxiom was omitted intentionally, since it is convenient to have a definitionwhich is valid for finite- as well as for infinite-dimensional spaces

In order to define the notion of the dimension, one needs the notion oflinear independence A finite or infinite sequence of vectors is calledlinearly independent if a relation such as = 0 implies = 0 for all n

If this is not the case we shall call the sequence linearly dependent

The maximal number of linearly independent vectors in t is called the

dimension of SW' Thus we say t has the dimension n = 1, 2, , if there

exists a set of n linearly independent elements in t but every set of (n + 1)

vectors is linearly dependent If n = cc, then we obtain what is usually

called the Hilbert space

An important question concerns the independence of the axioms If the

dimension n c cc, then Axioms 3 and 4 are consequences of the others, but not for n = cc.

If {L} is a linearly independent sequence, then the set of all elements

of the form

E

is an example of a linear manifold The elements of a linear manifold satisfy

the axioms in groups 1 and 2 but not necessarily Axioms 3 and 4 The

number n of elements in the set {1} is called the dinwnsion of the linearmanifold A formal definition and a more systematic discussion of linear

manifolds will be given in Section 2-4

www.pdfgrip.com

2-2 COMMENTS ON THE AXIOMS 21

If S is an arbitrary set of vectors in 3t', we may consider the smallest

linear manifold containing S Such a linear manifold always exists and is

unique We call this the linear manifold spanned by S

The positive definiteness of the scalar product implies the important

inequality of Schwartz:

(f, MfM MgM.

The proof of this is obtained immediately from the remark that the quantity

is nonnegative for all complex numbers cc, and for 0 its

minimum is equal tot

MfM2 — Kf,gN2

One also sees from this proof that the equality sign holds if and only if f

and g are linearly dependent

An easy corollary is Minkow ski's inequality (also called the triangle

inequality):

Hilbert space is not only a vector space but also a topological space.

This means we have a notion of convergence (or equivalently the notion of

open subsets) In order to define convergence we may use either the norm

or the scalar product When using the norm, we say a converges

in the norm to f if

n —.

In this case we speak of strong convergence

If we use the scalar product we may define a weak convergence as follows:The sequence converges weakly towards f if, for every g,

lim (ft, g) = (f, g).

n -.

The two kinds of convergence coalesce in finite dimensions, but not in infinite dimensions A sequence which converges weakly toward a limit

need not converge strongly toward anything (Problem 10)

Two vectors f and g with (f, g) = 0 are called orthogonaL A sequence

of vectors are called orthonormal if they satisfy

=

(We shall always denote vectors of norm 1 with Greek letters.)

t Cf Problcm 7.

22 THE AXIOMS OF HILBERT SPACE 2-2

For any vector f and any orthonormal sequence, Bessel's inequality

(Problem 11) is valid for any f:

v= 1

The orthonormal system is called complete if Bessel's inequality is

in fact an equality for any In that case one finds (Problem 11) that the

partial sums f,, converge strongly towards f, so that onemay write

f =EOPv,f)Qr

Such a system q,, is called a coordinate system The existence of coordinatesystems is an important consequence of separability (Axiom 3)

PROBLEMS

1 The complex numbers are a ililbert space of dimension 1.

2 The set of all square matrices A of n rows and columns (n < t) makes up aHilbert space of dimension n2, if the scalar product is defined by (A, B) =

Tr A*B, where Tr denotes the trace (sum of diagonal matrix elements).

3 In any Hilbert space one has the "law of the parallelogram":

lIf+ gil2 + If— gil2 = 2l]fl]2 + 2JJg112.

(Jordan and von Neumann [4].) A normed vector space permits the definition

of a scalar product such that (f, f) = lJf 112 if and only if the norm satisfies the

parallelogram law.

5 The vectors of a linear manifold satisfy the axioms in groups 1 and 2.

6 The intersection of any family LII, (1 e 1) of linear manifolds is again a linear

manifold.

7 For g 0 the minimum value of iif+ xgl]2 as runs through the complex

numbers is

f 2_ Kf,g)Phg 112

*8. (Generalization of the inequality of Schwartz.) Let with v = 1, ,

be a finite set of vectors; then the determinant of Gram Det (f',, 0 over, the equality sign holds if and only if the are linearly dependent.

More-9 Minkowski's inequality llf+ gil if ii + is a consequence of Schwartz's

inequality.

10 Any infinite orthonormal sequence of vectors ip,, converges weakly to zero No

such sequence can converge strongly to a limit.

11 For any orthonormal system {p.} one has the inequality

www.pdfgrip.com

2-3 REALIZATIONS OF HILBERT SPACE 23

The axiomatic which we have described in the preceding two sections stitutes a definition of abstract Hubert space The abstract space is a most

con-convenient notion when we wish to study very general properties which arenot related to any particular realization of the space in terms of other mathe-matical objects However, this is not the way Hilbert space appears in

physics. In concrete physical problems involving quantum mechanics,

Hilbert space appears always in a particular realization, for instance, as a

function space or as a space of sequences of numbers

Many realizations of Hilbert space are possible Such realizations are

also useful from a purely mathematical point of view, since they demonstrate

that the axioms of Section 2-1 are consistent We shall discuss only the

infinite dimensional space here, since the finite dimensional case is already

understood

The first of these realizations is the space It consists of all infinite

sequences of complex numbers f = with the property

C cc.

If A is a complex number, we define Af = and if g = is another

and (f, g)

n1

One easily verifies the axioms of groups 1 and 2, but the proofs of the

com-pleteness and separability theorems require certain technical devices

A second realization of Hilbert space is the function space L2(S, M, it)introduced at the end of Chapter 1 The elements of this space are classes

of equivalent functions in L2(S, M, p), where two functions f and g are

declared equivalent if

J If— gl2dp =

The functions f and g are then equal a.e with respect to the measure p If

f = {f(x)}, (x e S), is a vector of this space, then we define Af ={Af(x)}

If g = {g(x)} is another vector of this space, then we define

f + g = {f(x) + g(x)} and (f, g) = Jf*g dp.

That with these operations we obtain a Hubert space is a nontrivial assertion

Axioms 1 and 2 are easy enough to verify (Problem 3) hut Axioms 3 and 4

are deep theorems [1]

24 THE AXIOMS OF HILBERT SPACE 2-4

If the measure p is discrete and concentrated at an infinite number of

points, we obtain a slight generalization of the space j2, consisting of all

sequences subject to the condition that

<

for a fixed sequence of positive numbers p,, If we choose for these numbers

= 1 we again obtain the space

From the abstract point of view, all these different realizations represent

the same abstract Hilbert space This abstract space is completely

deter-mined by the axioms Thus if we have two different realizations, then there

exists a one-to-one mapping of one onto the other which preserves the

Hilbert-space structure Two realizations which stand in this relation to

one another are said to be isomorphic Two Hilbert spaces of the same

dimensions are always isomorphic

PROBLEMS

1 The space (2 isa linear vector space which satisfies the axioms of groups 1 and 2.

*2 The space (2 is separable and complete.

3 The function space L2(S, M, satisfies axioms 1 and 2 if scalar multiplication

addition and scalar product are defined by

Af= {Af(x)},

f+g {f(x) +g(x)},(f,g)

We shall now discuss with greater care a notion already introduced in

Section 2-2, the linear manifold

A subset 4' of a Hilbert space t is called a linear man (fold if f e 11

implies that Afe and iffe and g eA' imply that (f+ g) cA' We

say the set is stable with respect to multiplication with scalars and vector

addition

A linear manifold automatically satisfies axioms 1, 2, and 3 The first

two are just the definition of linear manifold, and the third is a consequence

of a theorem in topology which says that the subset of a separable set is

also separable But what about axiom 4?

Let us examine this question by means of an example. Consider the

space(2 It is easy to verify (Problem 1) that all sequences with only a finite

www.pdfgrip.com

2-4 LINEAR MANIFOLDS AND SUBSPACES 25

number of components # 0 are a linear manifold in j2 which is not

com-plete (Problem 2)

A vector f e t is a limit vector of A' if there exists a sequence e j7

such that f,, -+ f If every limit vector of A' belongs to 4', we call a

closed linear man jfold M, or a subspace A subspace is a Hilbert space.

Every linear manifold 4' can be closed by adding to it all the limit vectors;

if we want to express this process we denote it by M = A and call it the

closure of 4' The closure of 4' is thus the smallest subspace which contains

If 92 is a set of vectors, we denote by bol the set of all vectors orthogonal

to all vectors of 9' Thus

6°' = {f: (f, g) = 0 for all g e b°}

It is easy to verify that 6°' is a subspace (Problem 4) Furthermore, if

and 6°2 are two subsets of t such that 9 c 6°2, then 6°f. Since4' c M 2 it follows that M' c 4" and therefore (Problem 8)

But Mis the smallest subspace containing A', and so we must have 411± = M.

For every infinite dimensional subspace M there are infinitely many different

linear manifolds which are dense in 4' (For physical applications the

sub-spaces are more important than the linear manifolds.)

Fig 2—I Geometrical interpretation of the

decomposition of a vector with respect to

a subspace.

A very important property is the decomposition of vector f with respect

to a subspace M: To every subspace M there belongs a unique positionf = f1 + f2 such thatf1 e M andf2 e M' The geometrical content

decom-of this theorem can be seen at a glance from Fig 2—1

These considerations can be generalized to more than one subspace.

Suppose that is a sequence of mutually orthogonal subspaces such

that = 0. We say then that the span the entire space We can

then decompose every vectorf in a unique manner as a sum

with

26 THE AXIOMS OF HILBERT SPACE 2-5

This infinite sum is to be interpreted in the sense of strong convergence,

3 The intersection of two subspaces is again a subspace.

4 The set = {f: (f, g) = 0 for all g C 92} is a subspace.

5 If M and M-'- are two orthogonal subspaces, then the vectors f of the form

+ e M,f2 eM-'-, are a subspace

6 The subspace of Problem 5 is the entire space.

7 IffeMandfeM-'-,thenf=O.

8 If Mis a subspace then M-' '- = M

2-5 THE LATTICE OF SUI3SPACES

While the preceding sections represent basic material of the theory of Hilbertspace, the topic of this section is selected primarily with a view to the physicalinterpretation

Let M and N be two subspaces Their set-theoretic intersection M n N

is also a subspace It is the largest subspace contained both in M and N

In a similar way we may define the smallest subspace containing both M

and N, and we denote it by M u N.

The two operations n and u have properties similar to the set-theoreticintersections and unions introduced in Chapter 1, but there are some im-

portant differences which we shall now examine

It is convenient to introduce sets of subspaces which are closed with respect to these two operations, intersection (n) and union (u). Such asystem is an example of a lattice, and we denote it by &

www.pdfgrip.com

2-5 THE LATTICE OF SUBSPACES 27

If we require in addition that 2 be closed even with respect to a ably infinite number of intersections and unions, and, furthermore, that with

count-M there is also count-M-'- in 2, then we obtain a complete, orthocomplemented

lattice. Since M n M-'- = 0 and 0-'- = .t, such a lattice always contains 0

and A formal definition and detailed discussion of this notion will be

presented in Section 5-3.

If M1 (1 e I, some index set) is a family of subspaces, we denote the

union and intersection of these subspaces by

and

The difference between this and the lattice of subsets of a set comes to lightwhen we consider mixed operations involving unions and intersections inone and the same formula

Fig 2—2 Three subspaces of a two- Fig 2—3 Illustration of the relation

dimensional space which do not satisfy (A n B) u (A n B') = A.

the distributive law.

Let us examine this in a very special case which displays the teristic features of the general situation We take a two-dimensional Hilbertspace t and choose two one-dimensional subspaces, M and M', for instance

charac-Let N be any one-dimensional subspace # M and M'; then we have

(see Fig 2—2)

but

We see therefore that the operations u and n do not always satisfy the

distributive law as they do in the case for sets (Problem 3)

it is of great importance to have a criterion which tells under what

conditions subspaces satisfy the distributive law fit is easy to see that forany pair of sets A and B we have an identity (Fit 2—3):

(AnB)u(AnB')=A

which can he obtained immediately from the distributive law